easy_quantum_group
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| where $(e_i)_{i=1}^N$ is the standard basis of $\C^N$ and where for all $i_1,\ldots,i_\ell\in N$ the symbol $\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ is $1$ if the kernel, i.e., the induced partition with $k$ upper and $\ell$ lower points, of $(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ refines $p$ and is $0$ otherwise. | where $(e_i)_{i=1}^N$ is the standard basis of $\C^N$ and where for all $i_1,\ldots,i_\ell\in N$ the symbol $\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ is $1$ if the kernel, i.e., the induced partition with $k$ upper and $\ell$ lower points, of $(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ refines $p$ and is $0$ otherwise. | ||
| - | ===== Systematization ===== | + | ===== Taxonomy ===== |
| + | There are several systems to divide the class of of easy orthogonal quantum groups into cases. Let $G\cong(G(G),u)$ be an easy orthogonal quantum group and let $\Cscr\subseteq\Pscr$ be the category of partitions generating its corepresentation category. We say that $G$ is | ||
| + | |||
| + | * Classical case distinction: | ||
| + | * **orthogonal** or **case** $O$: if $\singleton\notin \Cscr$ and $\fourpart\notin \Cscr$, | ||
| + | * **bistochastic** or **case** $B$: if $\singleton\in \Cscr$ and $\fourpart\notin \Cscr$, | ||
| + | * **symmetric** or **case** $S$: if $\singleton\in \Cscr$ and $\fourpart\in \Cscr$, | ||
| + | * **hyperoctahedral** or **case** $H$: if $\singleton\notin \Cscr$ and $\fourpart\in \Cscr$, | ||
| + | * Liberty distinction: | ||
| + | * **free** or **liberated**: if all partitions of $\Cscr$ are [[category_of_all_non-crossing_partitions|non-crossing]], | ||
| + | * **half-liberated**: if $\Pabcabc\in \Cscr$ and $\Pabab\notin\Cscr$, | ||
| + | * **classical** or **commutative** or **group case**: if $\Pabab\in\Cscr$, | ||
| + | * Group theoreticity distinction: | ||
| + | * **group-theoretical**: if $\Paabaab\in\Cscr$, | ||
| + | * **non-group-theoretical**: if $\Paabaab\notin\Cscr$. | ||
| ===== Classification ===== | ===== Classification ===== | ||
easy_quantum_group.1580979039.txt.gz ยท Last modified: 2021/11/23 11:56 (external edit)
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